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By C. R. Wylie Jr., Mathematics
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What is the axis of the transformation? 22. Work Exercise 21, given that the image of the point (0,2) is the point (0,). 23. In Exercise 21, if the image of the point (0,a) is the point (0,b), determine the relation between a and b which will ensure that (0,a) is also the image of (0,b). 24. In Exercise 9, let L be the point in which the line through O containing a general point P and its image P′ intersects l. Prove that (OL)(PP′)/(OP)(PL) is a constant independent of P. Is this true if L is the point in which the line OP meets the vanishing line?
On the other hand, if PG is parallel to l, we can first choose a point, G1 such that G1G is not parallel to l, then use the construction we have just described to find the image, G′1 of G1 and finally determine the image of P by using the pair (G1, G′1) in place of the pair (G,G′). The transformation defined by the preceding construction is known as plane perspective. The line of invariant points, l, is known as the axis of the transformation, and the invariant point, O, is known as the center of the transformation.
The image of the circle Γ is therefore a hyperbola whose asymptotes are the images of the tangents to Γ at V1 and V2 (Fig. 13). Intermediate between the case in which the circle Γ intersects the vanishing line in two points and the case in which it does not intersect the vanishing line is the case in which it is tangent to the vanishing line, say at the point V. Then, with the exception of the vanishing line, which has no image, the lines which pass through V are transformed into lines, parallel to OV, each of which intersects the image of Γ in a single point.